Time-decay estimates for the linearized water wave type equations
Tesfahun A.
March 2022Birkhauser
Journal of Evolution Equations
2022#22Issue 1
Recently, A. Bulut showed that the free waves Sα(t) f: = exp (it| ∇ | α) f in 1D for α∈ (1 / 3 , 1 / 2] , which are known to be associated with the linearized gravity water wave equations, decay at time scale of order | t| - 1 / 2 for large t, provided that the Hx1(R)-norm of f and the Lx2(R)-norm of x∂xf are bounded. In this note we derive a decay estimate of order (1 - α) - 1 / 2(α| t|) -d/2 on Sα(t) f for all α∈ (0 , 1) and d≥ 1 , assuming a bound only on the B˙1,1d(1-α/2)(Rd)-norm of f. Our estimate extends to any dimension, a wider range of α and describes well the behaviour of the decay near α= 0 and α= 1 , without requiring a spatial-decay assumption on f or its derivative.
Fractional wave equations , Linearized water waves , Time-decay estimates
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Department of Mathematics, Nazarbayev University, Qabanbai Batyr Avenue 53, Nur-Sultan, 010000, Kazakhstan
Department of Mathematics
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